The city is located along the A2, and there traffic jams quite often between Hannover and Braunschweig.
The A39 (which connects the A2 with the A7) is now completed and the gap between Wolfenbüttel and Wolfsburg near Braunschweig is closed. This may change the traffic situation in southern Braunschweig a little bit in a positive way, there could be very good new hitchhiking spots towards east, south and west in the southern city. Just check it out!

The gas stations Rüningen Nord and Süd have been closed and new gas stations a bit further away from the city have been opened, Salzhitter Hüttenblick (as of Aug. 2018).

Hitchhiking out

From the city center take the Bus 560 from Braunschweig Rathaus to Wipshausen and exit in Zweidorf - Peiner Straße. From there walk the Rothbergstraße until you see the service station, place yourself there if you want to go east. Use the tunnel of the Rothbergstraße to cross the motorway and place yourself on this side of the service station if you want to go west. If the main gate of the service station is locked, you will have to either climb the fence or walk on the bridge and walk on the motorway till the station.

Comment: I lived around Braunschweig for 5 years and I never used the spot above. I think the Bus ride is to long and also I wouldn't climb the fence and definitely would not recommend to walk the A2 because it is one of Germany's major motorways and a little dangerous and absolutely forbidden. Also it is not difficult to get rides at the following alternatives.

Alternatives:

The ramp Braunschweig Ost: From the city center take the tram to Volkmarode. Then take the 6km walk to the ramp (or try to hitchhike to there). It is an average position (average 40 min), but my standard if go to Berlin because i often get direct rides from there.

Take the Bus to Groß Schwülper and exit near the ramp Braunschweig Hafen or take the walk from Hansestraße (3km) if you get stuck there. It is a good spot to go to the west (Hannover), east seems to be difficult. The spot is used by many vehicles from the industrial area Braunschweig Hafen, thats also why there is almost no traffic on sunday! Only good to hitch on working days!

Take Tram 1 to Braunschweig-Hansestrasse Ost or bus 484 to Braunschweig-Hansestrasse Ost or Daimlerstrasse. On the opposite side of the street from the McDonalds, you can wait on the corner with a sign for SZ (Salzgitter) which will bring you to the newly opened (Sept 2018) gas station "Salzgitter Hüttenblick). A lot of people drive from Braunschweig to at least Salzgitter. The spot is great because there is a traffic light for cars waiting to turn, and you can walk up and down the line of cars with your sign. Be ready to get into a car quickly, though, in case the light changes. I've never waited more than 5 minutes here.

At Salzgitter Hüttenblick, you'll find people going south on the A7, towards Kassel, but also going to Hannover.

Take the tram to Volkmarode and then try to hitchhike the B248, which leads to Wolfsburg.

Hitching In

from A2

Don't let yourself be dropped at Braunschweig Flughafen (Airport). It is a research airport and there is no traffic on weekends or after 6pm. From Braunschweig Ost it is a 6km walk to the next tram station (but maybe hitchable).
'Braunschweig Nord is a good place to be dropped off when you're coming from the A2, especially coming from the east. Exit 55 BS Kreuz Nord, make sure you slow down to not end up driving back onto the Autobahn (not very clearly marked). Then exit "Hansestrasse" and there you can drive into the McDonalds parking lot (make sure you're on the left-turn lane to be able to drive straight into the parking lot). This is a 2 km exit so just make sure you ask your driver. For him/her, it's easy from there to get back onto the highway. It's 3 km down Hansestr going west, so for someone driving towards Berlin it's a detour but someone driving west, it's in their direction anyway.
From there, you can take Tram 1 from Braunschweig-Hansestrasse Ost or bus 484 from Braunschweig-Hansestrasse Ost or Daimlerstrasse.

If you're coming from the 391, you can also be dropped off at BS Kreuz Nord.

Best is Braunschweig Hafen but it is still a walk 3km to the next tram-station. If coming from the west, it is possible that your driver leaves at BS Hafen and then takes you to Hansestraße (where you will find the tram station) your driver can easily go back on the motorway from there because there is another ramp.

A391

You can use all the ramps Weststadt, Lehndorf or Hansestraße, you will be able to find a bus from there.

A39

Since the construction of motorway A39 is finished, drivers prefer to take this road instead of A391.

From Braunschweig Südstadt (Exit 12) it's 1.5 km to Braunschweig Central Station. The bus needs five minutes. Your driver can turn at the petrol station to go back to the motorway.

It looks like it's possible to get dropped off at Kreuz Braunschweig Süd if your driver turns south towards Braunschweig Heidberg, then stops at the petrol station to let you get out and then turns right to a roundabout to turn. You have to walk further south to get to the tram station HEH-Kliniken.